2025/01/07 by Bhunia, Pintu, Damase, Sujit Sakharam, Khare, Apoorva
#15A60 #26C10 (secondary) #46M05 #47A12 #47A30 (primary) #47A63 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2501.03638
Suppose A=[aij]∈ Mn(ℂ) is a complex n × n matrix and B∈ B(H) is a bounded linear operator on a complex Hilbert space H. We show that w(A⊗ B)≤ w(C), where w(⋅) denotes the numerical radius and C=[cij] with cij= w(\beginbmatrix 0& aij aji&0 \endbmatrix ⊗ B). This refines Holbrook's classical bound w(A⊗ B)≤ w(A) ‖B‖ [J. Reine Angew. Math. 1969], when all entries of A are non-negative. If moreover aii≠ 0 ∀ i, we prove that w(A⊗ B)= w(A) ‖B‖ if and only if w(B)=‖B‖. We then extend these and other results to the more general setting of semi-Hilbertian spaces induced by a positive operator. In the reverse direction, we also specialize these results to Kronecker products and hence to Schur/entrywise products, of matrices: (1)(a) We first provide an alternate proof (using w(A)) of a result of Goldberg-Zwas [Linear Algebra Appl. 1974] that if the spectral norm of A equals its spectral radius, then each Jordan block for each maximum-modulus eigenvalue must be 1 × 1 ("partial diagonalizability"). (b) Using our approach, we further show given m ≥ 1 that w(A∘ m)≤ wm(A) - we also characterize when equality holds here. (2) We provide upper and lower bounds for the ℓp operator norm and the numerical radius of A⊗ B for all A ∈ Mn(ℂ), which become equal when restricted to doubly stochastic matrices A. Finally, using these bounds we obtain an improved estimation for the roots of an arbitrary complex polynomial.